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K614

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X(30), X(74), X(399), X(1138), X(1511)

reflection of X(1138) in X(1511)

K614 ia central focal cubic with center and focus the midpoint X(1511) of X(3)X(110) and X(74)X(399).

K614 and its isogonal transform generate a pencil of cubics, stable under isogonal conjugation, that contains the Neuberg cubic K001 and two decomposed cubics namely :

  • the union of the line at infinity and the circumconic (C) passing through X(74), X(399), X(1138), X(1272), X(1989).
  • the union of the circumcircle and the line (L) passing through X(30), X(146), X(323), X(399), X(1138).

This pencil also contains one isogonal circular nK with focus X(74) and, more generally, the singular focus of any cubic of the pencil is a point on the line X(3), X(110) = focus of K001.